don’t know why but if i try to explain why i don’t get this paradox the post gets removed
In: 0
[https://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox](https://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox)
>Have you ever noticed how sometimes what seems logical turns out to be proved false with a little math? For instance, how many people do you think it would take to survey, on average, to find two people who share the same birthday? Due to probability, sometimes an event is more likely to occur than we believe it to. In this case, if you survey a random group of just 23 people there is actually about a 50–50 chance that two of them will have the same birthday. This is known as the birthday paradox.
We can work out the odds that nobody in a group shares a birthday – i.e. they all have different birthdays.
If we ignore February 29th (which is a complication that makes very little difference), then:
* All possible dates work for person 1. Probability that it works is therefore 1
* For person 2, they cannot have the same birthday as person 1. There are therefore 364 days that work out of the 365 in a year. Odds of this are 364/365. Total odds are 1 * 364/365.
* For person 3, they cannot have the same birthday as person 1 or person 2. There are therefore 363 days that work out of the 365 in a year. Odds of this are 363/365. Total odds are 1* 364/365 * 363/365. We can refactorise this as (365*364*363)/365^(3)
* I’m sure you can spot the pattern here. For n people, we have (365*364*…*(366-n))/365^(n)
* For n=23, this number is less than 0.5 (it is 0.493), meaning that it is more likely than not that two people do in fact share a birthday for groups of 23 or more people.
Someone else did the actual math but an easier way to think about it is that the last person, number 23, has a 22/365 chance (or just over 1/20) to have the same birthday as one of the others and that’s IF all the others already have unique birthdays.
To explain why our intuitions lead us astray here, we’re used to thinking about coincidences like this from our own perspective, or at least from a single person’s perspective. When *you* walk into that room and ask a random person what their birthday is, yeah, chances are it’s not going to be the same as yours. Neither is the next person you ask, or the next.
But the question isn’t about any one specific person sharing your birthday, it’s about *any pair* of people. And even with only 23 people in the room, there are a lot of different pairs of people there – 253 of them. A 1 in 365 chance is small, but getting 253 shots at it…
ELI5 version:
It only takes 23 people before two people probably share a birthday.
This sounds weird, but it’s because you’re comparing Person 1’s birthday to Person 2, and Person 3, and Person 4, etc…and then *also* comparing Person 2’s birthday to Person 3, Person 4, etc…and then *also* comparing Person 3’s birthday to Person 4, Person 5, etc…
The number of comparisons you’re doing is way, way, *way* more than 23.
If you draw a flat sided shape and draw a line between each point. A triangle has 3 lines. A square has 6 lines. A pentagon has 10 lines. Think of the corners as people and the lines as comparing birthdays. The comparisons grow exponentially with the people just like the lines grow exponentially with the corners of a shape.
A 23 sided shape has a LOT of lines. But there’s only 365 possible birthdays.
The birthday paradox seems to defy our intuition.
If we consider that there is 365 days in a year, it seems crazy that in a room of 23+ people, some pair of people probably shares a birthday. That’s because 365 seems way larger than 23.
But in 23 people, how many pairs of people are there? Each pair is 2 people, so there’s 23 choices for the first person and 22 choices for the second person. Order doesn’t matter, so divide by 2 to avoid double counting.
23*22/2= 253 different pairs of people possible in the room. That means there’s effectively 253 different chances for a pair of people to share a birthday! Now it’s a lot easier to see why it’s so probable. Each chance is very unlikely, but 253 is so many chances that it adds up.
Let’s add the peoples one by one. For each person you add, you add a small % of chance of getting a double birthday.
So if you enter a room of 22 peoples, one of those two are true:
1. There was already a double birthday
2. There was no double birthday, and you have 20/365 = 6% of having the same birthday than someone else there.
And sure, that 6% is not much, **but that 6% is “just for you”**. There was already 22 peoples in the room, all of them with their own probability of having the same birthday as someone else.
And sure, the computation is not as easy as “23 times 6%” because probabilities are more difficult than that to compute (there are correlations and stuff).
But the idea is the same: if everyone on the room has a small probability of something happening, and that those are truly “different events” (and not like “the probability of dying from a nuclear war” where everyone dies or live at once), the room has a big probability of that thing happening.
Don’t think of it as *you* sharing a birthday with another person. Think of it as any two people from a large group sharing a birthday.
If Mrs Johnson’s 3rd grade class has 23 kids in it, then there’s a 50% chance two of the kids could have the same birthday. Not a specific two kids, just any two of them.
Latest Answers